In algebraic topology, a branch of mathematics, a simple space is a connected

topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...

that has a homotopy type of a

CW complex A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This cla ...

and whose

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

is abelian and acts trivially on the homotopy and homology of the universal covering space. Though not all authors include the assumption on the homotopy type.

Examples

Topological groups

For example, any

topological group In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two str ...

is a simple space (provided it satisfies the condition on the homotopy type).

Eilenberg-Maclane spaces

Most Eilenberg-Maclane spaces

K(A,n)

are simple since the only nontrivial homotopy group is in degree

n

. This means the only non-simple spaces are

K(G,1)

for

G

nonabelian.

Universal covers

Every connected topological space

X

has an associated (universal) simple space from the universal cover

\pi:U_X \to X

; indeed,

\pi_1(U_X) = *

and the universal cover is its own universal cover.

References

Dennis Sullivan Dennis Parnell Sullivan (born February 12, 1941) is an American mathematician known for his work in algebraic topology, geometric topology, and dynamical systems. He holds the Albert Einstein Chair at the City University of New York Graduate Ce ...

,
Geometric Topology
' Algebraic topology {{topology-stub